| L(s) = 1 | + 1.48·2-s − 3-s + 0.215·4-s + 4.37·5-s − 1.48·6-s − 4.26·7-s − 2.65·8-s + 9-s + 6.50·10-s + 2.03·11-s − 0.215·12-s − 1.63·13-s − 6.34·14-s − 4.37·15-s − 4.38·16-s − 17-s + 1.48·18-s + 6.68·19-s + 0.943·20-s + 4.26·21-s + 3.02·22-s + 0.998·23-s + 2.65·24-s + 14.1·25-s − 2.43·26-s − 27-s − 0.919·28-s + ⋯ |
| L(s) = 1 | + 1.05·2-s − 0.577·3-s + 0.107·4-s + 1.95·5-s − 0.607·6-s − 1.61·7-s − 0.939·8-s + 0.333·9-s + 2.05·10-s + 0.613·11-s − 0.0622·12-s − 0.453·13-s − 1.69·14-s − 1.12·15-s − 1.09·16-s − 0.242·17-s + 0.350·18-s + 1.53·19-s + 0.210·20-s + 0.930·21-s + 0.645·22-s + 0.208·23-s + 0.542·24-s + 2.82·25-s − 0.476·26-s − 0.192·27-s − 0.173·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.898592790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.898592790\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 5 | \( 1 - 4.37T + 5T^{2} \) |
| 7 | \( 1 + 4.26T + 7T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 13 | \( 1 + 1.63T + 13T^{2} \) |
| 19 | \( 1 - 6.68T + 19T^{2} \) |
| 23 | \( 1 - 0.998T + 23T^{2} \) |
| 29 | \( 1 + 5.55T + 29T^{2} \) |
| 31 | \( 1 - 4.45T + 31T^{2} \) |
| 37 | \( 1 + 9.01T + 37T^{2} \) |
| 41 | \( 1 - 4.65T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 5.93T + 47T^{2} \) |
| 53 | \( 1 + 1.83T + 53T^{2} \) |
| 59 | \( 1 - 9.08T + 59T^{2} \) |
| 61 | \( 1 - 2.34T + 61T^{2} \) |
| 67 | \( 1 - 6.16T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 3.88T + 73T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.852524987695691159695336453303, −7.11276549964155762679788161800, −6.70460560383525273253610057614, −5.88920293315905448361704339775, −5.65690085191399515521086628762, −4.93576998915969512443217556201, −3.85891711859737609415594435940, −3.03283965139200413986652214058, −2.27218229754398757571879489441, −0.857800750055788107386272357740,
0.857800750055788107386272357740, 2.27218229754398757571879489441, 3.03283965139200413986652214058, 3.85891711859737609415594435940, 4.93576998915969512443217556201, 5.65690085191399515521086628762, 5.88920293315905448361704339775, 6.70460560383525273253610057614, 7.11276549964155762679788161800, 8.852524987695691159695336453303
